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\begin{document}

{\hfill
\parbox{40mm}{{ICEN-PS-01/02} \\ {\ptoday} \vspace{4mm}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{{\LARGE {\rm Application of the Microlocal Analysis\\[3mm] to a Superfield 
Model in Superspace}}} 

\vspace{7mm}

{\large Daniel H.T. Franco} 

\vspace{0,5cm}

{Universidade Cat\'olica de Petr\'opolis -- (UCP) \\Grupo de F\'{\i}sica 
Te\'orica -- (GFT) \\[0,5mm]Rua Bar\~ao do Amazonas 124 - CEP:25685-070 - 
Petr\'opolis - RJ - Brasil.} 

\vspace{0,5cm} 

{\tt e-mail: daniel@gft.ucp.br}

\end{center}

\begin{abstract}
In this paper, we apply the microlocal analysis to study the 
singularity structure of the two-point function in a supersymmetric
model formulated in superspace language. 
\end{abstract}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%\hspace\parindent
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

There are topics, in the physical literature, which do not exhaust 
themselves, but deserve always new analyzes. Amongst these, the 
short-distance singularities (the notorious ultraviolet divergences) have a 
significant part. It is well-known that quantum field theories are deeply 
connected to the presence of these divergences. Although the 
renormalization program can overcome this problem in a mathematically 
proper way, there exists the need for a comprehension of the structure of 
these singularities. The suitable mathematical framework for this is the 
wavefront set, introduced by H\"ormander and Duistermaat~\cite{Hor1, DH} in 
the seventies for their analysis on the propagation of singularities of 
pseudodifferential operators. 

This subject is of growing importance, with a range of applications going 
beyond the original problems of linear partial equations. In particular, 
the  link with quantum field theories on a curved spacetime is now firmly 
established. A short time ago Radzikowski~\cite{Rad}, using the notion of 
wavefront set of a distribution instead of its singular support (which 
enables to eliminate the difference between local and global results), has 
generalized a conjecture by Kay~\cite{Kay} that the local Hadamard 
condition implies the global Hadamard condition. His proof one rely on a 
general wavefront set spectrum condition for the two-points distribution. 
Hadamard states are thought to be good candidates for describing physical 
states, at least for free quantum field theories in curved spacetime, since 
the work of De Witt and Brehme~\cite{WiBr} (see~\cite{Ful, KaWa, Wal} for a 
general review and references). Thereafter there has appeared a 
considerable amount of papers devoted to this problem~\cite{Ko}-\cite{Ver}. 

At the same time, it seems that not so much attention has been drawn to 
supersymmetric theories in this direction. Supersymmetry is a subject of 
considerable interest amongst physicists and mathematicians. It is not only 
it fascinanting in its own right, but even if 25 years have gone by after 
its proposal, there exists until a belief that it may play a fundamental 
role in particle physics. Calculations and phenomenological analysis of 
supersymmetry models are well-justified in view of the forthcoming 
generation of machines (NLC and LHC) which shall reveal some of the 
predicted supersymmetry particles, such as neutralinos, sleptons and may be 
indirectly squarks. It also has proven to be a tool to link the quantum 
field theory and non-commutative geometry~\cite{Wi, Jaf}. Hence, an 
extension of the technique of wavefront set applied for ordinary quantum 
field theories to supersymmetric ones seems desirable. 

In this work, we will devote special attention to the analysis of the 
singularity structure of two-point function to a superfield model, 
characterized in terms of the its wavefront set. Our analysis will be made 
directly in superspace~\cite{SaSt}. Elements of superspace are called 
supercoordinates which consist of the usual Minkowski spacetime coordinates  
and anticommuting Grassmann numbers. The concept of superspace was soon 
realized to represent the appropriate device for a formulation of 
supersymmetric field theories. 

An immediate advantage of the use of superfields is that it renders 
supersymmetry inherently manifest. Once one knows the action of the 
supersymmetry transformations in terms of the superspace coordinates, they 
systematically lead to the transformation laws for the components fields. A  
further advantage is that superfields automatically accomodate together 
with the physical fields (those associated with propagating degrees of 
fredom), a number of unphysical fields, the so-called auxiliary and 
compensating fields, which play a fundamental role in the formulation of 
both classical and second quantized supersymmetric field theories. 

The organization of this paper is as follows. After this Introduction, we 
present our free toy model in Section 2. We have confined our attention to 
an $N=1/2$-supersymmetric scalar model in two-dimensions for simplicity. In 
Section 3, we introduce the notion of the  microlocal analysis. Section 4 
is concerned with a study of the singularity structure to our superfield. 
In Section 5, we make some concluding remarks and comment on our future 
perspectives. Finally, the appendix contains some  properties of the model 
which has been employed here. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The Free Toy Model}
%$N=1/2$-Supersymmetric Scalar Model
\label{Sec1}
%\hspace\parindent
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

For the sake of simplicity in the presentation, we restrict our discussion 
to an $N=1/2$-supersymmetric scalar model in two-dimensions formulated over 
the $N=1/2$-superspace parametrized by coordinates $(x^{++}, x^{--}, 
\xi_+)$,\footnote{$x^{++}$ and $x^{--}$ are the light-cone coordinates and 
$\xi_+$ is a Grassmann coordinate (see appendix).} subjected to the motion 
equation 
\begin{equation}
\partial_{++}d_- \Phi(x,\xi)=0\,\,,
\label{Sec1.eq1}
\end{equation}
which is derivable from the free action
\begin{equation}
S_{\rm free}=\int d^2xd\xi\,\,\partial_{++}\Phi(x,\xi)d_-\Phi(x,\xi)\,\,.
\label{Sec1.eq2}
\end{equation}
The superfield, $\Phi(x,\xi)$, can be defined in terms of a power expansion  
of the spinorial variable $\xi$ with $x$-dependent coefficients, the 
so-called component fields: 
\begin{equation}
\Phi(x,\xi)=\phi(x)+i \xi_+ \psi_-(x)\,\,,
\label{Sec1.eq3}
\end{equation}
where $\phi(x)$ is a free boson field and $\psi(x)$ is a free Majorana-Weyl
spinor; both massless due to the chiral symmetry. As usual,
\begin{equation}
d_-=\partial_\xi-i \xi_+ \partial_{--}\,\,,
\label{Sec1.eq4}
\end{equation}
is a supersymmetric covariant derivative.

To our classical superfield, we may associate a quantum superfield, an 
operator-valued ``superdistribution,'' smeared with a ``supertest'' 
function defined by~\cite{Ost} 
\begin{equation}
F_+(x,\xi)=f_+(x)+i \xi_+ f(x)\,\,,\qquad F(x,\xi)\in {\cal D}({\Bbb 
R}^{2,1})= {\cal D}({\Bbb R}^{2})\otimes \Lambda C^1\,\,, 
\label{Sec1.eq5}
\end{equation}
where $\Lambda C^1$ stands for the 1-dimensional space parametrized by the 
Grassmann coordinate $\xi_+$. $f(x)\in {\cal D}({\Bbb R}^{2})$ is a scalar 
test function, while $f_+(x)\in {\cal D}({\Bbb R}^{2})$ is a spinorial test 
function. 

\,\,\,{\sf Remark:} The invariance under the Lorentz ``charge'' requires 
that the supertest functions have spinorial character. We refer to Apendix 
for more details on conventions and notations. 

\,\,\,For all $F(x,\xi),G(x,\xi)\in {\cal D}({\Bbb R}^{2,1})$, we define the
commutation relation
\begin{align}
\left[\Phi(F),\Phi(G)\right]=
\int d^2xd^2x^\prime d\xi d\xi^\prime\,\,
\Delta^{\rm susy}(x,\xi;x^\prime,\xi^\prime)
F(x,\xi) G(x^\prime,\xi^\prime)
\,\,.
\label{Sec1.eq6}
\end{align}
We call $\Delta^{\rm susy}(x,\xi;x^\prime,\xi^\prime)$ the Pauli-Jordan 
superdistribution, fundamental solution of the operator $\partial_{++}d_-$. 

\begin{proposition} The two-point function 
$\Delta^{\rm susy}(x,\xi;x^\prime,\xi^\prime)$ has the following form
\begin{equation}
\Delta^{\rm susy}(x,\xi;x^\prime,\xi^\prime)= 
d_-\left(\Delta(x-x^\prime)\delta(\xi_+-\xi^\prime_+)\right)\,\,, 
\label{Sec1.eq7}
\end{equation}
where $\Delta(x-x^\prime)$ is the fundamental solution of the Klein-Gordon 
operator. 
\label{pro1}
\end{proposition}

{\em Proof.} We first observe that the $\delta$-function over the Grassmann 
variable is defined by
\[
\delta(\xi_+-\xi^\prime_+)=\xi_+-\xi^\prime_+\,\,,
\]
which vanishes for $\xi_+=\xi^\prime_+$.

Now, using Eqs.(\ref{Sec1.eq4}) and (\ref{Sec1.eq5}), and then integrating 
over the $\xi_+$- and $\xi_+^\prime$-variables in the Eq.(\ref{Sec1.eq6}), 
we shall get the familiar result in components
\begin{align*}
\left[\Phi(F),\Phi(G)\right]=&
\int d^2xd^2x^\prime\,\,\left\{\Delta(x-x^\prime)f(x)g(x^\prime)-
i \partial_{--}\Delta(x-x^\prime)f_+(x)g_+(x^\prime)\right\}
\nonumber \\[3mm]
=&[\varphi(f),\varphi(g)]+\{\psi_-(f_+),\psi_-(g_+)\}
\,\,.
\end{align*}
where
\[
[\varphi(f),\varphi(g)]=(f,Eg)\,\,,\quad 
\{\psi_-(f_+),\psi_-(g_+)\}=(f_+,Sg_+) 
\,\,,
\]
for all $f,g,f_+,g_+ \in {\cal D}({\Bbb R}^{2})$. $E(x,x^\prime)\equiv 
\Delta(x-x^\prime)$ is the difference between the advanced and retarded 
fundamental solution of the Klein-Gordon operator, and $S(x,x^\prime)\equiv 
-i \partial_{--}\Delta(x-x^\prime)$ is the fundamental solution of the 
Dirac operator. This completes the proof. $\cqd$ 

\,\,\,Note in particular that due to the proposition \ref{pro1}, we get
\[
{\mbox {\rm supp}}\,\Delta^{\rm susy}(x,\xi;x^\prime,\xi^\prime) 
\subset {\mbox {\rm supp}}\,\Delta(x-x^\prime) 
\cup \{\xi_+ \not= \xi_+^\prime\} 
\,\,,
\] 
with ${\mbox {\rm supp}}\,\Delta(x-x^\prime)\subset 
\overline{V}_{+}\cup \left(-\overline{V}_{+}\right)$,
where $\overline{V}_{+}=\left\{x\in {\Bbb R}^2\,|\,x^2\geq 0,\,x^0\geq 
0\right\}$ is the future light-cone, being $V_+$ its interior.

Because we interpret $\Phi$ as an operator-valued superdistribution for 
every supertest function $F(x,\xi)\in {\cal D}({\Bbb R}^{2,1})$, the field 
equation then may be cast as below: 
\begin{equation}
\Phi(\partial_{++}d_-F)=0\,\,,
\label{Sec1.eq8}
\end{equation}
and due to the (\ref{Sec1.eq6}), we get that $\Delta^{\rm 
susy}(x,\xi;x^\prime,\xi^\prime) \in {\cal D}^\prime({\Bbb R}^{2,1})$ is a 
fundamental solution which solves the equation 
\begin{equation}
\Delta^{\rm susy}(\partial_{++}d_-F)=0\,\,.
\label{Sec1.eq9}
\end{equation}

The vacuum expectation value of the product $\Phi(F)\Phi(G)$ satisfies the 
relation 
\begin{align}
\left(\Omega,\Phi(F)\Phi(G)\Omega\right)=
\left(w_2^{\rm susy}(x,\xi;x^\prime,\xi^\prime),\,F(x,\xi) 
G(x^\prime,\xi^\prime)\right)\,\,. 
\end{align}
The distribution $w_2^{\rm susy}(x,\xi;x^\prime,\xi^\prime)$ extends the 
Wightman formalism. For this reason, we call $w_2^{\rm 
susy}(x,\xi;x^\prime,\xi^\prime)$ Wightman superdistribution of two-points.

As a consequence of the proposition \ref{pro1}, we obtain
\begin{equation}
w_2^{\rm susy}(x,\xi;x^\prime,\xi^\prime)= 
d_-\left(w_2(x-x^\prime)\delta(\xi_+-\xi^\prime_+)\right)\,\,, 
\label{Sec1.eq10}
\end{equation}
where $w_2(x-x^\prime)=\frac 1i\Delta^{\dagger}(x-x^\prime)$, with
\begin{equation}
\Delta^{\dagger}(x-x^\prime) =\frac i{2\pi}\int
d^2k\,\delta(k^2) \theta(k^0) e^{-\,ik\left( x-y\right)}\,\,.  
\label{two-point} 
\end{equation}

The Wightman superdistribution of $n$-points will be symbolically written 
under the form~\cite{Ost} 
\begin{equation}
w_n^{\rm susy}\left(x_1,\xi_1;\ldots;x_n,\xi_n\right) =\left(\Omega,\Phi 
\left(x_1,\xi_1\right) \ldots \Phi \left(x_n,\xi_n\right) \Omega\right)
\,\,, \label{wightman1}
\end{equation}
and
\begin{equation}
w_n^{\rm susy}\left(F_n\right)=\int \prod_{i=1}^nd^2x_i \prod_{i=1}^n 
d\xi_i\,\, w_n^{\rm susy}\left(x_1,\xi_1;\ldots;x_n,\xi_n\right) 
F_n\left(x_1,\xi_1;\ldots;x_n,\xi_n\right)\,\,. \label{wightman2} 
\end{equation}
In this definition, we have fixed the order in which we take the 
distribution and the test function. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Briefing on the Microlocal Analysis}
%\hspace\parindent
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The contents of this section can be found in refs.~\cite{RS2}--\cite{AS}. 
We shall introduce the mathematical tool necessary to investigate the 
distribution singularities, i.e., the wavefront set (${\cal{WF}}$) of a 
distribution, a refined description of the singularity spectrum. The 
main reason for using this tool is that it not only  describes the 
wavefront set of a distribution is singular, but also localize the momenta 
which constitute these singularities, yielding a simple characterization 
for the existence of distribution products. Similar notion was developed in 
other versions by Sato~\cite{Sa}, Iagolnitzer~\cite{Ia} and 
Sj\"ostrand~\cite{Sj}. The definition as known nowadays is due to 
H\"ormander. He used this terminology due to an existing analogy between 
his studies on the ``propagation'' of singularities and the classical 
construction of propagating waves by Huyghens. 

In the classical theory of propagating waves developed by Huyghens, the 
wave are propagated, for every instant, in a normal direction to the 
wavefront. In analogy with this theory, for a distribution $u$ we introduce 
its wavefront set ${\cal{WF}}(u)$ as subset on the momenta space. This 
subset consists of the points $(x,k)$ for which the direction of the  
vector $k$ is singular for $u$ in the point $x$. ${\cal{WF}}(u)$ is 
independent of the coordinate system chosen, and can be described locally. 

The most import point of the H\"ormander and Duistermaat analysis, also 
called microlocal analysis, is to transfer the study of singularities of 
distributions of the configuration space to the momenta space. For this, we 
need to ``localize'' the distribution on the neighborhood of the 
singularity, examining the result in the Fourier space. The technique 
consists in multiplying a distribution $u$ for a smooth function $\phi$ 
with support contained in a region $V$, with $\phi(x)\not=0$, for all $x 
\in V$. The distribution $\phi u$ can then be seen as a distribution of 
compact support on ${\Bbb R}^n$. From this point of view, all development 
is local in the sense that only the behaviour of the distribution on the 
arbitrarily small neighborhood of the singular point, in the configuration 
space, is relevant. 

As well-known~\cite{RS2, Hor2} a distribution of compact support, $u 
\in {\cal E}^\prime({\Bbb R}^n)$, is a smooth function if, and only if, its 
Fourier transform, $\widehat{u}$, rapidly decreases at infinity. By a fast 
decay at infinity, one must understanding that for all positive integer $N$ 
exists a constant $C_N$ such that 
\begin{equation}
|\widehat{u}(k)| \leq C_N (1+|k|)^{-N} < \infty\,, 
\qquad \forall\,N \in {\Bbb N};\,\,k \in {\Bbb R}^n\,\,. 
\label{Pepe} 
\end{equation}
If, however, $u \in {\cal E}^\prime({\Bbb R}^n)$ is not smooth, then the 
directions along which $\widehat{u}$ does not fall off sufficiently fast 
may be adopted to characterize the singularities of $u$.

Even though a distribution does not have compact support, still we can verify 
if its Fourier transform rapidly decreases in a given region $V$, going 
again through the technique of localization. Its Fourier transform will 
be defined as a distribution on ${\Bbb R}^n$, and will satisfy the property 
(\ref{Pepe}). 

\begin{definition}
Let $u \in {\cal D}^\prime({\Bbb R}^n)$ be a distribution and $\phi \in 
C_0^\infty(V)$ a smooth function with support $V \subset {\Bbb R}^n$. Then, 
$\phi u$ has compact support. 
\end{definition}

The Fourier transform of $\phi u$ produces a smooth function in the momenta 
space. 

\begin{lemma}
Consider $u \in {\cal D}^\prime({\Bbb R}^n)$ and $\phi \in C_0^\infty(V)$. Then
\[
\widehat{\phi u}(k)=u(\phi e^{-ikx})\,\,.
\]
Moreover, the restriction of $u$ to $V \subset {\Bbb R}^n$ is 
asymptotically limited for $k 
\rightarrow \infty$ if, and only if, for every $\phi \in C_0^\infty(V)$ 
exists a constant  $C_{\phi, N}$, such that 
\[
|\widehat{\phi u}(k)| \leq C_{\phi, N}(1+|k|)^{-N}<\infty\,, 
\qquad \forall\,N \in {\Bbb N};\,\,k \in {\Bbb R}^n\,\,. 
\]
\end{lemma}

If $u \in {\cal D}^\prime({\Bbb R}^n)$ is singular in $x$, and $\phi \in 
C_0^\infty(V)$ is $\phi(x)\not= 0$; then $\phi u$ is singular in $x$ and 
has compact support. In some directions $\widehat{\phi u}$ until will be 
asymptotically limited. This is called the set of {\em regular directions} 
of $u$. 

\begin{definition} Consider $u(x) \in {\cal D}^\prime({\Bbb R}^{n})$. 
The pair $(x,k) \in {\Bbb R}^n \times ({\Bbb R}^n\backslash 0)$ is called a 
point describing a regular direction for high momenta to $u(x)$ if, and 
only if, there exists a neighborhood $V$ of $x$, a conic neighborhood  $M$ of 
$k$, and a function $\phi(x) \in C_0^\infty(V)$, with $\phi(x)\not=0$, such 
that the Fourier transform of $\phi u$ is asymptotically limited for $k 
\rightarrow \infty$, i.e., 
\begin{equation}
|\widehat{\phi u}(k)| \le C_{\phi, N}(1+|k|)^{-N}<\infty \qquad 
\forall N \in {\Bbb N};\,\,k \in {\Bbb R}^{n}\,\,,
\end{equation}
where $C_{\phi, N}$ are constants. The wavefront set 
${\cal{WF}}(u)$ of the distribution $u(x)$ consists of the pairs $(x,k) 
\in {\Bbb R}^n \times ({\Bbb R}^n\backslash 0)$, of points $x$ in the
configuration space and $k$ in the Fourier space, such that the Fourier 
transform $\widehat{\phi u}$ not decay sufficiently rapid along of the 
direction $k$, for $|k| \rightarrow \infty$.
\end{definition}

The wavefront set ${\cal{WF}}(u)$ is conic in the sense that it remains 
invariant under the action of dilatations, i.e.
when we multiply the second variable by a positive scalar. This means that
if $k \in {\cal{WF}}(u)$ then $\lambda k \in {\cal{WF}}(u)$ for all 
$\lambda > 0$. Thus, $(x,k)$ is a point describing a regular direction if 
its ``localization'' $\phi u$ in a small neighborhood of $x$ has Fourier 
transform decreasing sufficiently fast for any power in a cone around $k$. 

From the definition of ${\cal{WF}}(u)$ for $u \in {\cal D}^\prime({\Bbb 
R}^{n})$ it follows that the projection $\pi_1({\cal{WF}}(u)) 
\rightarrow x$ is the singular support of $u$. Roughly speaking, if 
$(x,k)\in {\cal{WF}}(u)$ then $x$ specifies the localization of a 
singularity of $u$ and $k$ its ``direction of propagation'' 
\[
{\mbox{\rm $u$ is singular in $x$}}\quad \Longleftrightarrow \quad
\exists\,k \in {\Bbb R}^n \backslash 0 \left.\right| (x,k) \in 
{\cal{WF}}(u)\,\,.
\] 

The projection onto the second variable is represents by 
$\pi_2({\cal{WF}}(u)) \rightarrow \Sigma(u)$, where $\Sigma(u)$ is defined 
to be the complement in ${\Bbb R}^n\backslash 0$ of the set of all $k \in 
{\Bbb R}^n\backslash 0$ for which there is an open conic neighborhood $M$ 
of $k$ such that $\widehat{\phi u}$ if of rapid decrease in $M$. 

We emphasize that, as the notion of the wavefront set applies to 
distributions, it can be used to theories which are formulated in terms 
of pointlike fields. In the naive perturbative scheme of quantum field 
theories, one encounters formal products of fields which are a priori not 
well-defined. The ultraviolet problems involved in defining the product 
of these fields, can be conveniently controlled by the so-called 
{\bf H\"ormander Criterium}: 

\begin{theorem}{\rm (Theorem IX.45 in~\cite{RS2})}
Let $u$ and $v$ be distributions. Suppose that  
\[
(x,0)\not\in {\cal{WF}}(u) \oplus {\cal{WF}}(v)=
\{(x,k_1+k_2)\left.\right|(x,k_1)\in {\cal{WF}}(u),
(x,k_2)\in {\cal{WF}}(v)\}\,\,.
\]
Then, the product $uv$ exists and 
\[
{\cal{WF}}(uv)\subset{\cal{WF}}(u)\cup{\cal{WF}}(v)\cup
({\cal{WF}}(u)\oplus{\cal{WF}}(v))\,\,. 
\]
\end{theorem}

Hence, the product of distributions $u$ e $v$ 
is well-defined, in $x$, if $u$, or $v$, or both distributions are 
regulars in $x$. If $u$ and $v$ are singulars in $x$, the product exists if 
the sum of second component of the ${\cal{WF}}(u)$ and ${\cal{WF}}(v)$ 
in $x$ is different of zero.

Another important general fact is that the multiplication with a smooth 
function and differentiation do not enlarge the wavefront set:
\begin{align}
&{\cal{WF}}(au)\subset{\cal{WF}}(u)\quad {\mbox{if $a \in C^\infty$}}
\,\,,\\
&{\cal{WF}}(Pu)\subset{\cal{WF}}(u)\quad {\mbox{if $P$ is any linear 
differential operator}}\,\,. 
\end{align} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Singularity Structure of the Two-Point \\ 
Function of the Superfield $\Phi(x,\xi)$}
%\hspace\parindent
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

To explore the notion of the microlocal analysis of singularities, we need 
a method to compute the wavefront set of a distribution. For this, we 
will go through another important tool developed by H\"ormander, the so-called 
Fourier Integral Distribution, or Oscillatory Integral, employed in the 
study of pseudodifferential operators. Pseudodifferential operators allow one 
to give an alternative and more natural definition of the wavefront set.
Here, following the presentation of 
ref.~\cite{AS}, we  shall use the stationary phase method, which appears in 
the development of the theory of pseudodifferential operators in order to 
find the asymptotic behaviour of an integral of the form $\int dx\,\,e^{i 
\lambda 
\varphi(x)}a(x)$, when $\lambda \rightarrow \infty$ and $\varphi$ has 
critical points. 

Pseudodifferential operators generalize linear differential operators with 
variable coefficients. If $p(x,D)=\sum_{|\alpha \leq m|} 
a_\alpha(x)D^\alpha_x$ is a differential operator with $x$-dependent 
coefficients, then 
\begin{align}
p(x,D)u(x)&=\frac{1}{(2 \pi)^n}p(x,D)\int_{{\Bbb R}^n} d^nk\,\,
e^{ikx}\widehat{u}(k) \nonumber \\[3mm]
&=\frac{1}{(2 \pi)^n}\int_{{\Bbb R}^n} d^nk\,\,p(x,k)e^{ikx}\widehat{u}(k)
\,\,,\label{opd}
\end{align}
where $u(x) \in {\cal D}({\Bbb R}^n)$, $\widehat{u}(k)$ is the Fourier 
transform, $p(x,k)=\sum_{|\alpha \leq m|}a_\alpha(x)k^\alpha$. Replacing 
$p(x,k)$ by appropriate functions, called {\em symbols}, we obtain a 
pseudodifferential operator. The symbols is nothing else, in this case, but 
the polynomial $p(x,k)$ obtained by substituting the variable $k_j$ for the 
partial differentiations $D_j$. 

\begin{definition}
For an open set $X \subset {\Bbb R}^n$, and $m,\rho,\delta$ real numbers, 
with $0<\rho \leq 1$ and $0 \leq \delta <1$; one define the symbol space 
$S^m_{\rho,\delta}(X \times {\Bbb R}^s)$, on $X \times {\Bbb R}^s$, of {\bf 
order} $m$ and {\bf type} $(\rho,\delta)$, as being the space of smooth 
functions $a(x,k)$, such that for any compact set $\Omega 
\subset X$, where the functions $a(x,k)$ taking their values, and 
multi-indices $\alpha \in {\Bbb N}^n, 
\beta \in {\Bbb N}^s$, exists
a constant $C_{\alpha,\beta,\Omega}$ such that
\begin{equation}
\left|D^\alpha_xD^\beta_k a(x,k)\right|\leq C_{\alpha,\beta,\Omega}
(1+|k|)^{m-\rho|\beta|+\delta|\alpha|} \quad \forall\,x \in \Omega; k \in 
{\Bbb R}^s 
\,\,.\label{simbol}
\end{equation}
\end{definition}

The better constants $C_{\alpha,\beta,\Omega}$, in (\ref{simbol}) are 
semi-norms 
\begin{equation}
\left\|a\right\|_{\alpha,\beta,\Omega}= \sup_{x \in \Omega;\,k \in {\Bbb R}^s}
(1+|k|)^{\rho|\beta|-\delta|\alpha|-m}\left|D^\alpha_x D^\beta_k a(x,k)\right|\,\,.
\end{equation} 

\begin{definition} Given a symbol $a(x,y,k)$ in 
$S^m_{\rho,\delta}(X \times X \times {\Bbb R}^s)$, where the variable $k 
\in {\Bbb R}^s$ is the dual of $x_i \in X$, the pseudodifferencial operator 
is a Fourier integral operator 
\begin{equation}
Au(x)=\frac{1}{(2 \pi)^n}\int d^nkd^ny\,\,
e^{ik(x-y)}a(x,y,k){u}(y) \quad \forall\,u \in {\cal D}(X)
\,\,.\label{oif}
\end{equation}
We denote by $L^m_{\rho,\delta}(X)$ the space of these operators, and we 
say that $A \in L^m_{\rho,\delta}(X)$ is of order $\leq m$ and of type 
$(\rho,\delta)$. 
\end{definition}

In the physical applications of interest, it is sufficient to pay attention 
to the subclass of symbols $S^m_{1,0}$ first studied by Kohn and 
Nirenberg~\cite{KoNi}. A polynomial with respect to $k$ of degree $m$, with 
constant coefficients is of course a symbol $S^m_{1,0}$.

\,\,\,{\sf Example.} The inverse of $(1-\Delta):{\cal S}({\Bbb R}^n)\rightarrow 
{\cal S}({\Bbb R}^n)$, where $\Delta=\frac{\partial^2}{\partial x^2_1}+ 
\cdots +\frac{\partial^2}{\partial x^2_n}$ is the Laplacian operator, is 
given by
\begin{equation}
Au(x)=(1-\Delta)^{-1}=\frac{1}{(2 \pi)^n}\int d^nk\,\, 
e^{ikx}\frac{1}{(1+k^2)}\widehat{u}(k)
\,\,,\label{oif0}
\end{equation}
with $k^2=\sum_{i=1}^n k_i^2$. Hence, $A \in L^{-2}_{1,0}({\Bbb R}^n)$. 

\,\,\,{\sf Example.} If $A$ is a differential operator of order $\leq m$ 
on $X \subset {\Bbb R}^n$ with smooth coefficients, then $A \in 
L^m_{1,0}(X)$. In fact, if $A=\sum_{|\alpha|\leq m}a_\alpha(x)D^\alpha_x$, 
with $a \in C^\infty(X)$, then by inverse Fourier transform, we obtain 
\begin{equation}
Au(x)=\frac{1}{(2 \pi)^n}\int d^nk\,\, e^{ikx}a(x,k)\widehat{u}(k)
=\frac{1}{(2 \pi)^n}\int d^nkd^ny\,\, e^{ik(x-y)}a(x,k){u}(y)
\,\,,\label{oif1}
\end{equation}
where $a(x,k)=\sum_{|\alpha|\leq m}a_\alpha(x)k^\alpha \in S^m_{1,0}(X 
\times {\Bbb R}^s)$. 

In the example above, the kernel of $A$ is given by an integral so-called 
{\em oscillatory integral} 
\begin{equation}
K_A(x,y)=\frac{1}{(2 \pi)^n}\int d^nk\,\, e^{ik(x-y)}a(x,k) 
\,\,.\label{oif2}
\end{equation}

\begin{definition}
The oscillatory integral -- or Fourier integral distribution -- on $X 
\times {\Bbb R}^s$ is formally written as 
\begin{equation}
I_\varphi(a)=\int dk\,\,e^{i \varphi(x,k)}a(x,k)\,\,,\label{inosc}
\end{equation} 
where $\varphi(x,k)$ is a {\bf phase function} and $a(x,k)$ is an 
asymptotic symbol. 
\end{definition}

An important example of an oscillatory integral is the integral
\[
\int_{{\Bbb R}^n} dk\,\,e^{-ikx}=\delta(x)(2 \pi)^n\,\,,
\]
which defines the Dirac's distribution $\delta$.

\begin{definition}
Let $X \subset {\Bbb R}^n$ be open and $\Gamma$ an open cone in $X 
\times {\Bbb R}^s \backslash 0$. This means that $\Gamma$ is 
invariant if the second component in ${\Bbb R}^s$ is multiplyed by positive 
scalars. We say that the function $\varphi(x,k)\in C^\infty(\Gamma)$ is  a 
phase function in $\Gamma$ if 
\begin{enumerate}

\item $\varphi$ is homogeneous of degree 1 in $k$, i.e.,
$\varphi(x,\lambda k)=\lambda \varphi(x,k)$ if $(x,k)\in 
\Gamma \quad \forall\,\,\lambda > 0$; 

\item ${\rm Im}\,\varphi(x,k) \geq 0$;

\item $d\varphi=\sum_{i=1}^n \frac{\partial \varphi}{\partial x_i}dx_i+
\sum_{i=j}^s \frac{\partial \varphi}{\partial k_j}dk_j \not= 0$, i.e., 
$\varphi$ has no critical points in $\Gamma$. This means that at every 
point in $\Gamma$, some $\frac{\partial 
\varphi}{\partial x_i}$ or $\frac{\partial \varphi}{\partial k_j}$ 
is non-vanishing.
\end{enumerate} 
\end{definition}

\begin{definition}
If $\varphi \in C^\infty(X \times {\Bbb R}^s \backslash 0)$ is a phase 
function, we call 
\[
{\cal C}_\varphi=\left\{(x,k) \in X \times {\Bbb R}^s \backslash 0
\left.\right| \varphi^\prime_k(x,k)=0 \right\}\,\,, 
\] 
the critical set of $\varphi$. Here, 
$\varphi^\prime_k(x,k)=\left(\frac{\partial \varphi}{\partial k_1},\ldots, 
\frac{\partial \varphi}{\partial k_s}\right)$. The manifold of stationary 
phase is the point set
\[
\Lambda_\varphi=\left\{(x,\varphi^\prime_x(x,k)) \left.\right| 
(x,k) \in {\cal C}_\varphi; k \not= 0 \right\}\,\,,
\] 
with, $\varphi^\prime_x(x,k)=\left(\frac{\partial \varphi}{\partial 
x_1},\ldots,\frac{\partial \varphi}{\partial x_n}\right)$.
\end{definition}

It is the behaviour of $a(x,k)$ and $\varphi(x,k)$ near ${\cal C}_\varphi$ 
which determines the singularities of $I_\varphi(a)$. 

\begin{lemma}{\rm(Lemma 3 in~\cite{RS2}, pg. 101)}
$\Lambda_\varphi$ is a closed subset of $(X \times {\Bbb R}^s 
\backslash 0)$ and if $(x,k) \in \Lambda_\varphi$, then 
$(x,\lambda k) \in \Lambda_\varphi$ for all $\lambda \in {\Bbb R}_+$.
\label{zaz}
\end{lemma}

\begin{proposition}
If $\varphi(x,k)$ is a phase function on $X \times {\Bbb R}^s 
\backslash 0$ and $a(x,k) \in S^m_{\rho,\delta}(X \times {\Bbb R}^s \backslash 
0)$, with $\delta < 1$, $\rho > 0$; then ${\cal {WF}}(I_\varphi(a)) 
\subset \Lambda_\varphi$.  
\label{dani}
\end{proposition}

Before proving proposition \ref{dani}, it is suitable we recall a few 
additional results. 

\begin{lemma}
Let $X \subset {\Bbb R}^n$ be an open set, and $u \in C^\infty_0(X)$. 
If $\varphi \in C^\infty(X)$ is a phase function such 
that ${\rm Im}\,\varphi \geq 0$ and $d\varphi 
\not= 0$, i.e., $\varphi$ has no critical points on the support
of $u$, then the integral 
\[
I(\lambda)=\int dx\,\,e^{i \lambda \varphi(x)}u(x)\,\,, 
\] 
rapidly decreases when $\lambda \rightarrow \infty$.
\end{lemma} 

The stationary phase method applies when $d\varphi$ is allowed to vanish, 
but instead one makes the hypothesis that all the critical points of 
$\varphi$ are nondegenerate which means that, if $d\varphi(x_0)=0$, the 
Hessian matrix of $\varphi$ at $x_0$, $                                                       
\left(\frac{\partial^2 \varphi(x)}{\partial x_j \partial x_k}\right)_
{1\leq j,k \leq n}$, is necessary nonsingular~\cite{Tre, AS}. Hence, if 
$\varphi \in C^\infty(X)$ is such that ${\rm Im}\,\varphi \geq 0$ and $u 
\in C^\infty_0(X)$, the asymptotic behaviour of $I(\lambda) 
\rightarrow \infty$ is determined by $\varphi$ and $u$, in the
neighborhood of the set of critical points of $\varphi$, i.e., when 
$d\varphi=0$. Thus, the {\em essential} contributions must always come from  
the points where the phase $\varphi$ is real and stationary.  

\,\,\,{\em Proof of the proposition \ref{dani}.} We first assume that $(x,k) \in
(X \times {\Bbb R}^s \backslash 0)\backslash \Lambda_\varphi$. Now, we 
consider $u \in C^\infty_0(\Omega)$, where $\Omega \subset X$ represents a 
compact set, with $u(x) \not= 0$ for all $x \in \Omega$. Then, the integral 
\[
\widehat{I_\varphi(au)}(p)= \int dk 
dx\,\,e^{i \widetilde{\varphi}(x,k,p)}a(x,k)u(x)\,\,,
\quad \widetilde{\varphi}(x,k,p)=\varphi(x,k)-xp\,\,, 
\] 
must rapidly decrease in the conic neighborhood $V$ of $k$ for all $p \in 
V$. In order to prove this, we apply the method of stationary phase. We put 
$p=\lambda p^\prime$ and perform the change of variables $k 
\rightarrow \lambda k^\prime$, such that with $p,k \in V$ also 
$p^\prime$ and $k^\prime$ are contained in $V$ for all $\lambda>0$. So we 
obtain 
\begin{align*}
\widehat{I_\varphi(au)}(\lambda p)=\lambda^{-n} \int dk dx\,\,
e^{i \widetilde\varphi(x,\lambda k,\lambda p)}a(x,\lambda k)u(x)\,\,,
\end{align*} 
dropping by convenience the $^\prime$.

Using the homogeneity of the phase function, then 
$\widetilde\varphi(x,\lambda k,\lambda p)=\lambda\widetilde\varphi(x,k,p)$ 
if $(x,k,p)$ belongs to an open cone $\Gamma$ in $(X \times {\Bbb R}^s 
\backslash 0)$ for all $\lambda > 0$. By Lemma 5 (pg.105) in~\cite{RS2}, 
there exists a differential operator, $L$, such that $^tL\,e^{i \lambda 
\widetilde\varphi}=e^{i \lambda \widetilde\varphi}$, (where $^tL$ is its 
adjoint) whose coefficients are homogeneous of degree $-1$ in $k,p$. This 
operator is given by 
\[
L=\frac{1}{i \lambda \Phi(x,k,p)} 
\left(\frac{\partial\varphi(x,k)}{\partial x_i} - p_i \right)D_{x_i}\,\,,
\quad \Phi(x,k,p)=\left(\frac{\partial\varphi(x,k)}{\partial x_i}
- p_i \right)^2\,\,. 
\] 
Consequently, we obtain 
\begin{align*}
\widehat{I_\varphi(au)}(\lambda p)=&\lambda^{-n} \int dk dx\,\,
(^tL)^\alpha\,e^{i \lambda \widetilde\varphi}a(x,\lambda k)u(x) \\[3mm] 
=&\lambda^{-n} \int dk dx\,\,e^{i \lambda 
\widetilde\varphi} 
\,L^\alpha\left(a(x,\lambda k)u(x)\right)
\,\,. 
\end{align*} 

Hence, we get the estimate:
\begin{align*}
\left|\widehat{I_\varphi(au)}\right|=&\left| \lambda^{-n-|\alpha|} \int dk 
dx\,\,\frac{i^{|\alpha|}e^{i\lambda(\widetilde\varphi(x,k,p))}}
{\Phi^\alpha(x,k,p)} \left(\frac{\partial\varphi(x,k)}{\partial x_i} - p_i 
\right)^\alpha D^\alpha_{x_i}\left(a(x,\lambda k)u(x)\right)\right| \\[3mm]
\leq& \sum_{|\alpha| \leq N}
C_{\alpha,\varphi,\Omega}\left(\sup_{x\in 
\Omega}\left|D^\alpha u(x)\right|\right) (1+|\lambda|)^{m-(1-\delta)|\alpha|-n} 
\,\,, 
\end{align*} 
which rapidly decreases if $m-n-(1-\delta)|\alpha|<0$ to $\lambda 
\rightarrow \infty$. By hypothesis, as $\delta < 1$ and since $|\alpha|$ 
can be made arbitrarily large, then $I_\varphi(au)$ is asymptotically 
limited for $\lambda 
\rightarrow \infty$ for an open cone in $(X \times {\Bbb R}^s 
\backslash 0)$. If $m$ can be chosen arbitrarily negative, $I_\varphi(au)$ 
is asymptotically limited even for $|\alpha|=0$. As a result of this, we 
deduce that $(x,k) \notin {\cal {WF}}(I_\varphi(a))$, which completes the 
proof. $\cqd$ 

\,\,\,We are finally ready to state our main result: 

\begin{proposition}
The two-points function $w^{\rm susy}_2(x,\xi;x^\prime,\xi^\prime)$ of the 
free massless superfield $\Phi(x,\xi)$ has its wavefront set given by: 
\[
{\cal{WF}}(w^{\rm susy}_2)\subset {\cal{WF}}(w_2)\,\,,
\] 
with
\begin{align*}
{\cal{WF}}(w_2)=&\left\{(x,k_1),(x^\prime,k_2)\in({\Bbb R}^2 
\times {\Bbb R}^2 
\backslash 0)\left.\right|x\not=x^\prime;(x-x^\prime)^2=0; 
k_1||(x-x^\prime);\right. \\[3mm] &\left. k_1+k_2=0;k_1^0\geq 0\right\}\cup 
\left\{(x,k_1),(x^\prime,k_2)\in({\Bbb R}^2 \times {\Bbb R}^2 
\backslash 0)\left.\right|x=x^\prime;\right. \\[3mm]
&\left. k_1+k_2=0;k_1^2=0; k_1^0\geq 0\right\}\,\,. 
\end{align*}
\label{wfs} 
\end{proposition} 

{\em Proof.} We first observe that by ``Ectoplasmic Integration Theorem'' 
of Gates~\cite{Gat}, the topology of a supermanifold must be generated 
essentially from its bosonic submanifold. So the Grassmannian sector of 
superspace cannot produce an effect on the singular structure of the 
two-points $w^{\rm susy}_2(x,\xi;x^\prime,\xi^\prime)$. Now, using the 
representation of $w^{\rm susy}_2(x,\xi;x^\prime,\xi^\prime)$, 
eq.(\ref{Sec1.eq10}), and exploring the fact that a differential operator 
decrease the wavefront set, we are allowed to conclude that
\begin{equation}
{\cal{WF}}(w^{\rm susy}_2)\subset {\cal{WF}}(w_2)\,\,.
\label{wfssusy}
\end{equation} 
The proof then follows that of the Theorem IX.48 of~\cite{RS2} with 
$w_2(x,x^\prime)$ given by following representation of the Fourier 
transform 
\[
\widehat{w_2}(x,x^\prime)=\frac{1}{2\pi}\delta(k_1+k_2)
\theta(k_1^0)\delta(k_1^2)\in {\cal D}^\prime({\Bbb R}^2 
\times {\Bbb R}^2)\,\,.
\] 
This completes the proof. $\cqd$

\,\,\,{\sf Remark:} The proposition~\ref{wfs} provides us with a ``global'' 
wavefront set. In our setting the word ``global'' means that the singular 
support of all component fields is embodied in eq.(\ref{wfssusy}). Moreover,
it reflects the fact that the fundamental solution of 
the operator $\partial_{++}d_-$ is singular on the light-cone. 

\,\,\,That the bosonic sector is responsible by carrying all singular 
structure of the superspace is not too surprising. Being the key idea of the 
H\"ormander and Duistermaat analysis the shift of the study of singularities 
of the configuration space to the Fourier space, we recall that by convention 
the Fourier transform of any object written in superspace language, it is 
realized only in bosonic sector of the superspace. Moreover, apparently, 
there no exist reason to have superspaces where the topological properties 
of the superspace are substantially different from the bosonic submanifold 
it contains, otherwise this should not be consistent with the usual 
components-by-projection technique. 

The singularity structure of the Feynman free superpropagator can be 
investigated in the same way. In fact, in theory of the free quantum 
superfield, one defines the Feynman free superpropagator by 
\[
i\Delta_F^{\rm susy}(x,\xi;x^\prime,\xi^\prime)= w^{\rm 
susy}_2(x,\xi;x^\prime,\xi^\prime)+ \Delta_{\rm ret}^{\rm 
susy}(x,\xi;x^\prime,\xi^\prime)\,\,. 
\]

To end up, it is worthwhile to emphasize that a generalization to a more 
general superfield is straightforward, up to possible (but surmountable) 
conceptual problems existing in microlocal analysis of a superfield whose 
component fields are of the multi-component type. In fact, in the case of a 
more general superfield model whose component fields contain spinorial 
and/or vectorial fields of the multi-component type, the ``global'' 
wavefront set does not contain any information about the components of the 
component fields that are singular. Hence, although the superspace methods 
display advantages over components approaches to supersymmetry, making 
quantum calculations easy, it seems desirable to go through the component 
expansions of superfield and superspace if we wish find out the singular 
components of spinorial and/or vectorial fields. This will require the 
notion of the polarization set ${\cal{WF}}_{\rm pol.}$, a refinement of the 
wavefront set, introduced by Dencker~\cite{Den} in order to analyze the 
singularities of multi-component distributions. For example, if $u \in 
{\cal D}^\prime(X,{\Bbb C}^N)$ is a multi-component distribution on $X$, 
taking their values in ${\Bbb C}^N$; then $u=u_j$, $j=1,\ldots,N$, where 
$u_j \in {\cal D}^\prime(X,{\Bbb C}^N)$, so that 
\begin{equation}
{\cal{WF}}(u)=\bigcup_{j=1}^n {\cal{WF}}(u_j)\,\,.
\end{equation}
As our ``global'' wavefront set, the wavefront set so defined does not 
contain any information about the components of the distribution $u$ that 
are singular. It is the Dencker polarization set which allows us to 
identify the singular components of the multi-component distribution $u$ 
(cf.~\cite{Holl, Kra, SaVe} for recent applications of this idea). 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Concluding Remarks and Outlook}
%\hspace\parindent
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The main purpose of this paper is an attempt to apply the 
microlocal analysis to study the singularity structure of the two-point 
function for a superfield model directly in superspace. As mentioned in 
the Introduction, the concept of superspace was soon realized to represent 
the appropriate device for a formulation of supersymmetric field theories. 
This has an advantage of rendering supersymmetry manifest and, moreover, 
the formalism accomodate together with physical fields, the auxiliary and 
compensating fields needed for the formulation of supersymmetric field 
theories. 

In spite of the hard works which have been made for a comprehension of the 
quantization of supersymmetric theories via the formalism of Feynman 
supergraphics and superpropagators\footnote{A comprehensive account of the 
quantum theory through the algebraic renormalization approach can be found 
in the textbook of Piguet and Sibold~\cite{PiSi}.}~\cite{Gri}, we think 
that the use of the microlocal analysis to the study of the singularity 
structure of the superpropagators might refine our understanding of the 
source of its divergences. This will may contribute significatively to 
better understanding of interacting supersymmetric quantum field theories. 

On the other hand, the inclusion of the gravitation in this scenario 
remains an open problem of Physics and an active area of current research. 
Although a significative progress in the energy scale has been reached, the 
Planck scale ($10^{19}Gev$) (at which effects from quantum gravity are 
expected to become important) remains unaccessible. From the purely 
theoretical point of view, all the attempts to include gravity in the 
quantization program failed up to now. Alternative proposals such as 
Supergravity, Kaluza-Klein~\cite{KaKl} and String theories~\cite{Str}, and 
more recently the D-brane theory~\cite{Bra} and the Baez-Rovelli 
formulation~\cite{Bae, Rov}, have elucidated the role of quantum gravity, 
without, however, providing conclusive results. For this reason, and 
because of relevant scale for the MSSM ($10^{3}Gev$), a reasonable 
approximation should be to consider the interaction of matter and 
gravitational fields as a quantum field theory in curved spacetimes. The 
gravitational field is included as a background field and the matter fields 
are quantized as operator-valued Wightman fields. This framework has a wide 
range of physical applicability, the most prominent being the gravitational 
effect of particle creation in the vicinity of black-holes, learned about 
for the first time by Hawking~\cite{Haw}. 

From an axiomatic point of view, whereas the most of the Wightman axioms 
can be implemented on a curved background spacetime ${\cal M}$, the 
spectrum condition (which expresses the positivity of the energy) 
represents a serious conceptual problem. While the Poincar\'e covariance, 
in particular the translations, guarantees the positivity of the spectrum, 
and fixes a unique vacuum state if ${\cal M}={\Bbb R}^4$ is the Minkowski 
space, this familiar concept of field theory does not exist in a generic 
curved background spacetime. So, in general, no useful notion of a vacuum 
state (or equivalently of a particle interpretation) exist, too. 

An advice on how to define the spectrum condition, at least for free 
quantum field theory in curved spacetime, was given by Wald~\cite{Wald} for  
purpose of finding the expectation value of the energy-momentum tensor. For 
free fields, this approach led to the concept of a Hadamard state. The 
import discovery by Radzikowski~\cite{Rad} that the global Hadamard 
condition can be locally characterized in terms of the wavefront set, has 
made the connection with the spectrum condition much more transparent. 
After this, considerable advances have been made in this direction, 
especially by Hamburg group. Motivated by insights from these recent 
advances, and as a very interesting matter for an investigation, we intend 
to understand how to describe the Hadamard condition directly in 
superspace. As a example, we investigate the Hadamard condition for the 
Wess-Zumino model~\cite{CDH1} (cf.~\cite{Ko} for an analysis in 
components). As a next step, we intend to study the renormalization of this 
model on a ``supercurved'' background~\cite{CDH2} a la 
Brunetti-Fredenhagen~\cite{BF}. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Acknowledgements}
%\hspace\parindent
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We would like to thank J.A. Helay\"el-Neto for helpful discussions and 
illuminating comments on supersymmetry. We are also grateful to the staff 
of the Group of Theoretical Physics at UCP for the warm hospitality and for 
the pleasant atmosphere. 

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\appendix 
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\section{Notations and Conventions}
\label{Apea}
%\hspace\parindent
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

For sake of completeness, we quote some properties of the model which has 
been employed here. This material is included to render the paper as 
self-contained as possible. 

In two-dimensional theories, a coordinate system much used is the light-cone 
coordinate one, defined by
\begin{equation}
x^{++}=\frac {x^0+x^1}{\sqrt{2}}\,\,,\qquad x^{--}=\frac 
{x^0-x^1}{\sqrt{2}}\,\,. 
\end{equation} 
Taking into account that the two-dimensional Minkowski space has a metric 
given by $\eta_{\mu\nu}={\rm diag.}(1,-1)$, one can show that the line 
element in light-cone coordinates assumes the form 
\begin{equation}
ds^2=2dx^{++}dx^{--}\,\,,
\end{equation} 
indicating which the metric tensor in this system is given by
\begin{equation}
\eta=
\begin{pmatrix}
0 & 1 \\ 1 & 0
\end{pmatrix}\,\,.
\label{eta}
\end{equation} 
Therefore, we get
\[
x^{++}=x_{--}\,\,,\qquad x^{--}=x_{++}\,\,.
\]

An immediate advantage of the use of the light-cone coordinates is that, 
they do not mix under Lorentz transformations:
\begin{equation}
\begin{pmatrix}
\tilde{x}^{++} \\ \tilde{x}^{--}
\end{pmatrix}=
\begin{pmatrix}
e^\alpha & 0 \\ 0 & e^{-\alpha}
\end{pmatrix}
\begin{pmatrix}
x^{++} \\ x^{--}
\end{pmatrix}
\,\,,
\end{equation}
where $\alpha$ is a parameter of $SO(1,1)$. We say that $x^{++}$ and 
$x^{--}$ have Lorentz ``charges'' $+1$ and $-1$, while that $x_{++}$ and 
$x_{--}$ have Lorentz ``charges'' $-1$ and $+1$, respectively. 

We denote the spinorial representation of the Lorentz group by:
\begin{equation}
\begin{pmatrix}
\tilde{\xi}^{+} \\ \tilde{\xi}^{-}
\end{pmatrix}=
\begin{pmatrix}
e^{\alpha/2} & 0 \\ 0 & e^{-\alpha/2}
\end{pmatrix}
\begin{pmatrix}
\xi^{+} \\ \xi^{-}
\end{pmatrix}
\,\,,
\end{equation}
such that the Dirac matrices, in the Majorana-Weyl representation, have the 
following form
\begin{equation}
\gamma_0=
\begin{pmatrix}
0 & 1 \\ -1 & 0
\end{pmatrix}\,\,,\qquad
\gamma_1=
\begin{pmatrix}
0 & 1 \\ 1 & 0
\end{pmatrix}\,\,.
\end{equation} 

To lower and rise spinor indices, we employ the following convention:
\begin{equation}
\xi^\alpha=\epsilon^{\alpha \beta}\xi_\beta\,\,,\qquad
\xi_\alpha=\epsilon_{\alpha \beta}\xi^\beta\,\,,
\end{equation}
where
\begin{equation}
\epsilon^{\alpha \beta}=
\epsilon_{\alpha \beta}=
\begin{pmatrix}
0 & 1 \\ -1 & 0
\end{pmatrix}\,\,.
\end{equation} 

Chiral spinors are always defined on spaces of even dimensions through a 
matrix, $\gamma_{d+1}$, such that 
\begin{equation}
\left(\gamma_{d+1}\right)^2=1\,\,,\qquad
\left\{\gamma_{d+1},\gamma_{\mu}\right\}=0\,\,.
\end{equation}
 
We define projection operators by:
\begin{equation}
P_{\pm}=\frac{1 \pm \gamma_{d+1}}{2}\,\,,
\end{equation}
such that chiral spinors are obtained by
\begin{equation}
\psi_{\pm}=\frac{1 \pm \gamma_{d+1}}{2}\psi\,\,,
\end{equation}
where $\psi_+$ and $\psi_-$ are independent spinors of Weyl. Majorana-Weyl 
spinors can only be defined in dimensions $d=2+8n$,
if there is a time-like dimension and $(D-1)$ space-like coordinates. 
They are chirals and satisfy the Majorana condition 
\begin{equation}
\psi^*=\psi\,\,.
\end{equation}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\end{document}
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